Related papers: The number of Hecke eigenvalues of same signs

The author gives nontrivial upper and lower bounds for the number of primes in the interval $[x - x^{\theta}, x]$ for some $0.52 \leqslant \theta \leqslant 0.525$, showing that the interval $[x - x^{0.52}, x]$ contains prime numbers for all…

The eigenvalue problem for the Laplacian on bounded, planar, convex domains with mixed boundary conditions is considered, where a Dirichlet boundary condition is imposed on a part of the boundary and a Neumann boundary condition on its…

For bound states of atoms and molecules of $N$ electrons we consider the corresponding $K$-particle reduced density matrices, $\Gamma^{(K)}$, for $1 \le K \le N-1$. Previously, eigenvalue bounds were obtained in the case of $K=1$ and…

We establish various small deviation inequalities for the extremal (soft edge) eigenvalues in the beta-Hermite and beta-Laguerre ensembles. In both settings, upper bounds on the variance of the largest eigenvalue of the anticipated order…

We consider the first positive Steklov eigenvalue on planar domains. First, we provide an example of a planar domain for which a first eigenfunction has a closed nodal line. Second, we establish a lower bound for the first positive…

Let G be a torsion-free finite-index subgroup of SL(n,Z) or GL(n,Z), and let d be the cohomological dimension of G. We present an algorithm to compute the eigenvalues of the Hecke operators on the integral cohomology of degree d-1 for n =…

We obtain new inequalities with alternating signs of H\"{o}lder and Minkowski type.

We address the optimal constants in the strong and the weak Stechkin inequalities, both in their discrete and continuous variants. These inequalities appear in the characterization of approximation spaces which arise from sparse…

Let $\pi$ be a Hecke cusp form for $\mathrm{SL}_3(\mathbb{Z})$. We bound the second moment average of $L(s,\pi)$ over a short interval to obtain the subconvexity estimate $$ L(1/2+it, \pi) \ll_{\pi, \varepsilon}…

We obtain a lower bound for \[ \#\{x/2< p_{n}\leq x:\ p_n \equiv\ldots\equiv p_{n+m}\equiv a\text{ (mod $q$)},\ p_{n+m} - p_{n}\leq y\}, \] where $p_{n}$ is the $n^{\text{th}}$ prime.

Let $f$ and $g$ be two Hecke-Maass cusp forms of weight zero for $SL_2(\mathbb Z)$ with Laplacian eigenvalues $\frac{1}{4}+u^2$ and $\frac{1}{4}+v^2$, respectively. Then both have real Fourier coefficients say, $\lambda_f(n)$ and…

We prove a Lichnerowicz type lower bound for the first nontrivial eigenvalue of the $p$-Laplacian on K\"ahler manifolds. Parallel to the $p = 2$ case, the first eigenvalue lower bound is improved by using a decomposition of the Hessian on…

In this paper we consider in a bounded domain $\Omega \subset \mathbb{R}^N$ with smooth boundary an eigenvalue problem for the negative $(p,q)$-Laplacian with a Steklov type boundary condition, where $p\in (1,\infty)$, $q\in (2,\infty)$ and…

In this paper, we consider lower order eigenvalues of Laplacian operator with any order in Euclidean domains. By choosing special rectangular coordinates, we obtain two estimates for lower order eigenvalues.

We introduce the biharmonic Steklov problem on differential forms by considering suitable boundary conditions. We characterize its smallest eigenvalue and prove elementary properties of the spectrum. We obtain various estimates for the…

We introduce the the fractional Laplacian on a subgraph of a graph with Dirichlet boundary condition. For a lattice graph, we prove the upper and lower estimates for the sum of the first $k$ Dirichlet eigenvalues of the fractional…

We consider the problem of inference on the signs of $n>1$ parameters. We aim to provide $1-\alpha$ post-hoc confidence bounds on the number of positive and negative (or non-positive) parameters. The guarantee is simultaneous, for all…

In this paper, we study the upper bounds for discrete Steklov eigenvalues on trees via geometric quantities. For a finite tree, we prove sharp upper bounds for the first nonzero Steklov eigenvalue by the reciprocal of the size of the…

For a bounded domain $\Omega$ with a piecewise smooth boundary in an $n$-dimensional Euclidean space $\mathbf{R}^{n}$, we study eigenvalues of the Dirichlet eigenvalue problem of the Laplacian. First we give a general inequality for…

We prove two explicit bounds for the multiplicities of Steklov eigenvalues $\sigma_k$ on compact surfaces with boundary. One of the bounds depends only on the genus of a surface and the index $k$ of an eigenvalue, while the other depends as…